Forward speed motions of a submerged body. The Cauchy problem. (English) Zbl 0568.35059

Hamdache, K.

Math. Methods Appl. Sci. 6, 371-392 (1984).

The linearized problem for the motion of a submerged body in a perfect, inviscid, irrotational fluid, occupying an unbounded domain of \(R^ 3\), moving forward with a forced motion depending on (x,y) but not z, is \(\Delta u=0\), \(u|_{\Sigma}=\phi\), \(\partial u/\partial n|_{\Gamma}=f(t,\cdot)\), and \(\phi_ t-a\cdot \phi_ x+w\cdot \phi_ y=-b\eta\), \(\eta_ t-a\cdot \eta_ x+(w\cdot \eta)_ y=du/\partial n|_{\Sigma},\) \(\phi |_{t=0}=\phi_ 0\), \(\eta |_{t=0}=\eta_ 0\). In this paper the elliptic and the initial- value problem are viewed as coupled by a pseudo-differential operator T. The Yosida approximation \(T^{\epsilon}\) of T leads to a regularized initial-value problem. The perturbation theory of nonlinear semigroups is used to obtain weak solutions of the problem as \(\epsilon\) goes to zero.

Reviewer: G.F.Webb

MSC:

35L45	Initial value problems for first-order hyperbolic systems
35J25	Boundary value problems for second-order elliptic equations
47H20	Semigroups of nonlinear operators
35D05	Existence of generalized solutions of PDE (MSC2000)

Keywords:

motion of a submerged body; perfect, inviscid, irrotational fluid; pseudo-differential operator; Yosida approximation; regularized initial- value problem; perturbation; nonlinear semigroups; weak solutions

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